Does Minimum of Complex Set Subset Exist?

[Bashyboy]

Homework Statement

The following doesn't come from a textbook, and I am very uncertain whether it is true or false. Suppose that ##B \subseteq \mathbb{C}## is a convex set, and consider the set ##L_B := \{|b|: b \in B \}##.

Homework Equations

The Attempt at a Solution

My question is, will ##min~L_B## exist? My thought was that ##B## being convex implied that ##L_B## is convex; but I am unsure whether convexity of ##L_B## is sufficient to conclude that ##min~L_B##. Please refrain from giving me an entire answer, but I would appreciate a few hints.

 

[micromass]

Think about an open set in ##\mathbb{R}##.

 

[Bashyboy]

Ah, a counterexample! For instance, if we have ##B = (0,1)##, then ##L_B = B##, yet ##B## does not have a minimum. What if we stipulate that ##B## must also be compact?

 

[micromass]

Does a compact set always have a minimum?

 

[HallsofIvy]

A compact set is both bounded and closed.

 

[STEMucator]

Since a compact set is bounded and closed, the infimum is the minimum, and the minimum exists.